The evolution of galaxy cluster counts is a powerful probe of several fundamental cosmological parameters. A number of recent studies using this probe have claimed tension with the cosmology preferred by the analysis of the Planck primary CMB data, in the sense that there are fewer clusters observed than predicted based on the primary CMB cosmology. One possible resolution to this problem is systematic errors in the absolute halo mass calibration in cluster studies, which is required to convert the standard theoretical prediction (the halo mass function) into counts as a function of the observable (e.g., X-ray luminosity, Sunyaev-Zel'dovich flux, optical richness). Here I propose an alternative strategy, which is to directly compare predicted and observed cluster counts as a function of the one-dimensional velocity dispersion of the cluster galaxies. I show that the velocity dispersion of groups/clusters can be theoretically predicted as robustly as mass but, unlike mass, it can also be directly observed, thus circumventing the main systematic bias in traditional cluster counts studies. With the aid of the BAHAMAS suite of cosmological hydrodynamical simulations, I demonstrate the potential of the velocity dispersion counts for discriminating even similar ΛCDM models. Then, I compare the abundance of groups in the GAMA survey to the predictions from BAHAMAS to constrain the values of several cosmological parameters. Additionally, I investigate the role of active galactic nuclei (AGN) in galaxy evolution. The color bimodality of galaxy populations roughly divides galaxies into two groups: blue, star-forming galaxies, and red, quiescent galaxies. One theory that explains how high-mass, red, non-star-forming galaxies maintain this condition is the duty cycle hypothesis. This hypothesis invokes AGN feedback from low luminosity radio-loud AGN (LERGs) to deposit mechanical heating into the intergalactic medium, thus preventing star formation. I test this hypothesis by comparing the half-light radii of quiescent elliptical galaxies with LERG host galaxies using a large multi-wavelength sample from two surveys, UKIDSS/UDS, and ULTRAVISTA/COSMOS. The radius distribution of the two groups are similar, thus providing evidence for the duty cycle hypothesis. I also check the star formation activity of the LERGs. For the duty cycle to hold, LERGs should reside within non-star-forming galaxies. However, I find that a subset of LERGs appear to be dusty star forming galaxies.